Superstatistics of Diatomic Molecules with the Shifted Deng-Fan Potential Model

In this article, we discuss the thermodynamic properties of the shifted Deng-Fan potential for HCl, CrH, CuLi, and ScF diatomic molecules using the q-deformed superstatistics approach. The partition function is obtained with the help of the generalized Boltzmann factor from the modified Dirac delta distribution. In addition, thermodynamic functions such as entropy, specific heat capacity, free energy, and mean energy are obtained using the partition function. Our results are presented graphically, and the ordinary statistical quantities are recovered when the deformation parameter tends to zero. Our results may be useful in the study of thermal fluctuations in atomic and molecular systems involving short-range interactions.

An Observation on Distribution of Prime Numbers

In this research first, a sequence of properties called delta is assigned to each prime number and then examined. Deltas are only dependent on the distribution of prime numbers, so the results obtained for the delta distribution can be considered as a proxy for the distribution of prime numbers. The first observation was that these properties are not unique and different prime numbers may have the same value of delta of a given order. It was found that a small number of deltas cover a large portion of prime numbers, so by recognizing repetitive deltas, the next prime numbers can be predicted with a certain probability, but the most important observation of this study is the normal distribution of deltas. This research has not tried to justify the obtained observations and instead of answering the questions, it seeks to ask the right question.

Generalization of the Winfree model to the high-dimensional sphere and its emergent dynamics

<p style='text-indent:20px;'>We present a high-dimensional Winfree model in this paper. The Winfree model is a mathematical model for synchronization on the unit circle. We generalize this model compare to the high-dimensional sphere and we call it the Winfree sphere model. We restricted the support of the influence function in the neighborhood of the attraction point to a small diameter to mimic the influence function as the Dirac delta distribution. We can obtain several new conditions of the complete phase-locking states for the identical Winfree sphere model from restricting the support of the influence function. We also prove the complete oscillator death(COD) state from the exponential <inline-formula><tex-math id="M1">\begin{document}$ \ell^1 $\end{document}</tex-math></inline-formula>-stability and the existence of the equilibrium solution.</p>

A priori error estimates of regularized elliptic problems

Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work we show a-priori rates of convergence of this approximation process in standard Sobolev norms, with minimal regularity assumptions on the approximation of the Dirac delta distribution. The application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp $$H^1$$ H 1 and $$L^2$$ L 2 error estimates for the corresponding regularized problem. As an application, we show how finite element approximations of a regularized immersed interface method results in the same rates of convergence of its non-regularized counterpart, provided that the support of the Dirac delta approximation is set to a multiple of the mesh size, at a fraction of the implementation complexity. Numerical experiments are provided to support our theories.

Identification of train loads from the dynamic responses of an integrated sleeper in situ

The monitoring of railway tracks can be performed through several measurement techniques. Recently, a method of diagnosing the railway track has been proposed using fiber Bragg gratings integrated inside the railway sleeper. An analytical model for the dynamics of railway sleepers has been developed allowing calculation of the sleeper responses. In this model, using the relation between the rail forces and displacements of a periodically supported beam, the sleeper’s dynamic equation is written with the help of the Euler–Bernoulli beam and Dirac’s delta distribution. Subsequently, the sleeper dynamic responses are calculated using the Green’s function. This article presents an application of this model to identify the train loads from the strains measured in situ. Based on this model, we can obtain a matrix which presents the link between the loads and the sleeper responses. Then, by substituting the Fourier transform of measured strains at the middle and at the two rail seats of the sleeper, the train loads can be quickly calculated by inverting the matrix with the help of MATLAB. This method is validated by the experiments. Numerical examples with the measurement in situ are presented to show identified wheel loads from experimental signals.

Realistic models for filling and abundance discrepancy factors in photoionized nebulae

ABSTRACT When comparing nebular electron densities derived from collisionally excited lines (CELs) to those estimated using the emission measure, significant discrepancies are common. The standard solution is to view nebulae as aggregates of dense regions of constant density in an otherwise empty void. This porosity is parametrized by a filling factor f &lt; 1. Similarly, abundance and temperature discrepancies between optical recombination lines (ORLs) and CELs are often explained by invoking a dual delta distribution of a dense, cool, metal-rich component immersed in a diffuse, warm, metal-poor plasma. In this paper, we examine the possibility that the observational diagnostics that lead to such discrepancies can be produced by a realistic distribution of density and temperature fluctuations, such as might arise in plasma turbulence. We produce simulated nebulae with density and temperature fluctuations described by various probability distribution functions (pdfs). Standard astronomical diagnostics are applied to these simulated observations to derive estimates of nebular densities, temperatures, and abundances. Our results show that for plausible density pdfs, the simulated observations lead to filling factors in the observed range. None of our simulations satisfactorily reproduce the abundance discrepancy factors (ADFs) in planetary nebulae, although there is possible consistency with H ii regions. Compared to the case of density-only and temperature-only fluctuations, a positive correlation between density and temperature reduces the filling factor and ADF (from optical CELs), whereas a negative correlation increases both, eventually causing the filling factor to exceed unity. This result suggests that real observations can provide constraints on the thermodynamics of small-scale fluctuations.